My Attempt at your Data

library(pander)
ydat <- read.csv("yDat.csv", header=TRUE)

My Final Guess

My final guess at your model.

\[ Y_i = \beta_0 + \beta_1 X_2 + \beta_2 X_4 + \beta_3 X_9 + \beta_4 X18 + \beta_5 X_2 X_{6==4} + \epsilon_i \]

with my 95% confidence interval estimates of the coefficients as follows.

final.lm <- lm(Y ~ X2 + X4 + X9 + X18 + X2:I(X6==4), data=ydat)
mytable <- round(confint(final.lm, level=1-0.05/length(final.lm$coef)), 2)
betas <- paste0("$\\beta_", 0:(length(final.lm$coef)-1), "$")
rownames(mytable) <- betas
colnames(mytable) <- c("Lower", "Upper")
pander(mytable)
  Lower Upper
\(\beta_0\) 1 1
\(\beta_1\) 1 1
\(\beta_2\) 1 1
\(\beta_3\) -7 -7
\(\beta_4\) 6 6
\(\beta_5\) 1.2 1.2

My Approach to Finding the Model

First Pairs Plot

Glance at the data.

X3 can be dropped because it is all a single value.

X19 can be dropped because it is related to X5, perfectly.

Otherwise, nice work. Nothing obvious to pick from. I stared for a while at this point trying to decide where to go next. Finally, I just went with makeTable9.3’s suggestions.

pairs(ydat)

ydat <- ydat[,-c(4,20)]

Here is what Table 9.3 suggested between the AIC and PRESS.

lm.1 <- lm(Y ~ X2 + X4 + X9 + X18, data=ydat)
summary(lm.1)
## 
## Call:
## lm(formula = Y ~ X2 + X4 + X9 + X18, data = ydat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.28810 -0.20252  0.00871  0.18863  1.89601 
## 
## Coefficients:
##              Estimate Std. Error  t value Pr(>|t|)    
## (Intercept)  0.729030   0.319165    2.284   0.0271 *  
## X2           1.700774   0.115179   14.766  < 2e-16 ***
## X4           1.003762   0.005864  171.184  < 2e-16 ***
## X9          -7.017835   0.042868 -163.710  < 2e-16 ***
## X18          5.973591   0.511507   11.678 3.24e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5231 on 45 degrees of freedom
## Multiple R-squared:  0.9985, Adjusted R-squared:  0.9983 
## F-statistic:  7372 on 4 and 45 DF,  p-value: < 2.2e-16
par(mfrow=c(1,2))
plot(lm.1, which=1:2)

This next plot shows some exciting leads in X2 and X6.

pairs(cbind(R=lm.1$res, ydat), col=as.factor(ydat$X6==4))

lm.2 <- lm(Y ~ X2 + X4 + X9 + X18 + X2:I(X6==4), data=ydat)
par(mfrow=c(1,2))
plot(lm.2, which=1:2)

pairs(cbind(R=lm.2$res, ydat), col=as.factor(ydat$X6==4))

Let’s look more closely at X16, is it all the same values? Nope. Still, it doesn’t seem to have anything to do with Y.

par(mfrow=c(1,2))
stripchart(ydat$X16, method="stack")
stripchart(1/ydat$X16)

plot(Y ~ X16, data=ydat, col=as.factor(X10))

Check my guess

lm.3 <- lm(Y ~ X2 + X4 + X9 + X18 + X2:I(X6==4), data=ydat)
summary(lm.3)
## 
## Call:
## lm(formula = Y ~ X2 + X4 + X9 + X18 + X2:I(X6 == 4), data = ydat)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -6.774e-13 -1.556e-13 -1.461e-14  8.823e-14  6.490e-13 
## 
## Coefficients:
##                     Estimate Std. Error    t value Pr(>|t|)    
## (Intercept)        1.000e+00  1.567e-13  6.383e+12   <2e-16 ***
## X2                 1.000e+00  7.573e-14  1.321e+13   <2e-16 ***
## X4                 1.000e+00  2.869e-15  3.486e+14   <2e-16 ***
## X9                -7.000e+00  2.092e-14 -3.347e+14   <2e-16 ***
## X18                6.000e+00  2.491e-13  2.409e+13   <2e-16 ***
## X2:I(X6 == 4)TRUE  1.200e+00  8.712e-14  1.377e+13   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.548e-13 on 44 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 2.49e+28 on 5 and 44 DF,  p-value: < 2.2e-16
par(mfrow=c(1,2))
plot(lm.3, which=1:2)

Table 9.3

First Make Table that gave me the leads

makeTable9.3(ydat)
##     p                                                           model
## 1   1                                                     (intercept)
## 2   2                                                             X12
## 3   3                                                           X4,X9
## 4   4                                                        X2,X4,X9
## 5   5                                                    X2,X4,X9,X18
## 6   6                                                 X2,X4,X6,X9,X18
## 7   7                                             X2,X4,X6,X9,X10,X18
## 8   8                                          X1,X2,X4,X6,X9,X10,X18
## 9   9                                      X1,X2,X4,X6,X9,X10,X12,X18
## 10 10                                  X1,X2,X4,X6,X9,X10,X12,X17,X18
## 11 11                              X1,X2,X4,X6,X9,X10,X12,X17,X18,X20
## 12 12                          X1,X2,X4,X6,X9,X10,X12,X13,X17,X18,X20
## 13 13                      X1,X2,X4,X6,X9,X10,X12,X13,X15,X17,X18,X20
## 14 14                  X1,X2,X4,X6,X9,X10,X12,X13,X15,X16,X17,X18,X20
## 15 15               X1,X2,X4,X6,X8,X9,X10,X11,X12,X13,X15,X17,X18,X20
## 16 16           X1,X2,X4,X6,X8,X9,X10,X11,X12,X13,X15,X16,X17,X18,X20
## 17 17       X1,X2,X4,X6,X8,X9,X10,X11,X12,X13,X14,X15,X16,X17,X18,X20
## 18 18    X1,X2,X4,X5,X6,X8,X9,X10,X11,X12,X13,X14,X15,X16,X17,X18,X20
## 19 19 X1,X2,X4,X5,X6,X7,X8,X9,X10,X11,X12,X13,X14,X15,X16,X17,X18,X20
##           SSEp       R2p       R2ap           Cp      AICp        SBCp
## 1  8081.257454 0.0000000 0.00000000           NA        NA          NA
## 2  7178.582333 0.1116998 0.09319358 3.284481e+04 396.23556    1.901768
## 3   103.709927 0.9871666 0.98662051 4.311779e+02 186.37259 -206.049177
## 4    49.633656 0.9938582 0.99345762 1.854114e+02 151.52616 -238.983589
## 5    12.313637 0.9984763 0.99834083 1.641859e+01  83.82807 -304.769657
## 6     8.440419 0.9989556 0.99883687 6.722932e-01  66.94430 -319.741402
## 7     8.155225 0.9989908 0.99885003 1.365593e+00  67.22565 -317.548033
## 8     7.701133 0.9990470 0.99888821 1.285035e+00  66.36108 -316.500580
## 9     7.469780 0.9990757 0.99889531 2.225020e+00  66.83598 -314.113655
## 10    7.250323 0.9991028 0.99890096 3.219513e+00  67.34500 -311.692608
## 11    7.046295 0.9991281 0.99890450 4.284700e+00  67.91780 -309.207786
## 12    6.958457 0.9991389 0.99888968 5.882244e+00  69.29059 -305.922971
## 13    6.908676 0.9991451 0.99886783 7.654156e+00  70.93160 -302.369938
## 14    6.875586 0.9991492 0.99884196 9.502542e+00  72.69154 -298.697975
## 15    6.823818 0.9991556 0.99881784 1.126535e+01  74.31366 -295.163835
## 16    6.783591 0.9991606 0.99879024 1.308104e+01  76.01803 -291.547438
## 17    6.767850 0.9991625 0.99875648 1.500892e+01  77.90188 -287.751573
## 18    6.766114 0.9991627 0.99871795 1.700097e+01  79.88905 -283.852376
## 19    6.765903 0.9991628 0.99867663 1.900000e+01  81.88749 -279.941913
##        PRESSp
## 1          NA
## 2  7860.93609
## 3   116.25231
## 4    69.67406
## 5    22.38044
## 6    16.37542
## 7    16.26006
## 8    15.94943
## 9    15.94167
## 10   16.43823
## 11   16.66825
## 12   17.67388
## 13   18.14183
## 14   19.68187
## 15   19.36895
## 16   22.05955
## 17   23.08059
## 18   24.60817
## 19   25.59408

Confirmation Make Table that Settled It

makeTable9.3(cbind(ydat, X6p=ydat$X6==4, X2.6=ydat$X2*(ydat$X6==4)))
##     p
## 1   1
## 2   2
## 3   3
## 4   4
## 5   5
## 6   6
## 7   7
## 8   8
## 9   9
## 10 10
## 11 11
## 12 12
## 13 13
## 14 14
## 15 15
## 16 16
## 17 17
## 18 18
## 19 19
## 20 20
## 21 21
##                                                                       model
## 1                                                               (intercept)
## 2                                                                       X12
## 3                                                                     X4,X9
## 4                                                                X4,X9,X2.6
## 5                                                            X4,X9,X18,X2.6
## 6                                                         X2,X4,X9,X18,X2.6
## 7                                                     X2,X4,X9,X16,X18,X2.6
## 8                                                 X2,X4,X9,X14,X16,X18,X2.6
## 9                                             X2,X4,X9,X14,X16,X18,X6p,X2.6
## 10                                        X2,X4,X9,X14,X15,X16,X18,X6p,X2.6
## 11                                     X2,X4,X6,X9,X14,X15,X16,X18,X6p,X2.6
## 12                                 X2,X4,X6,X9,X10,X14,X15,X16,X18,X6p,X2.6
## 13                             X2,X4,X6,X9,X10,X14,X15,X16,X18,X20,X6p,X2.6
## 14                         X2,X4,X6,X9,X10,X12,X13,X14,X15,X16,X18,X6p,X2.6
## 15                      X1,X2,X4,X8,X9,X10,X12,X13,X14,X16,X17,X18,X20,X2.6
## 16                X2,X4,X9,X10,X11,X12,X13,X14,X15,X16,X17,X18,X20,X6p,X2.6
## 17             X2,X4,X6,X9,X10,X11,X12,X13,X14,X15,X16,X17,X18,X20,X6p,X2.6
## 18          X2,X4,X6,X7,X9,X10,X11,X12,X13,X14,X15,X16,X17,X18,X20,X6p,X2.6
## 19        X1,X2,X4,X6,X7,X8,X9,X10,X12,X13,X14,X15,X16,X17,X18,X20,X6p,X2.6
## 20     X1,X2,X4,X5,X6,X7,X8,X9,X10,X12,X13,X14,X15,X16,X17,X18,X20,X6p,X2.6
## 21 X1,X2,X4,X5,X6,X7,X8,X9,X10,X11,X12,X13,X14,X15,X16,X17,X18,X20,X6p,X2.6
##            SSEp       R2p       R2ap            Cp        AICp
## 1  8.081257e+03 0.0000000 0.00000000            NA          NA
## 2  7.178582e+03 0.1116998 0.09319358  1.202796e+29   396.23556
## 3  1.037099e+02 0.9871666 0.98662051  1.737695e+27   186.37259
## 4  4.733204e+01 0.9941430 0.99376101  7.930644e+26   149.15207
## 5  1.131705e+01 0.9985996 0.99847511  1.896210e+26    79.60821
## 6  3.013845e-24 1.0000000 1.00000000  1.249800e+01 -2750.34624
## 7  2.738680e-24 1.0000000 1.00000000  9.887523e+00 -2753.10735
## 8  2.320681e-24 1.0000000 1.00000000  4.883806e+00 -2757.05081
## 9  2.121337e-24 1.0000000 1.00000000  3.543726e+00 -2756.25958
## 10 2.033311e-24 1.0000000 1.00000000  4.068822e+00 -2755.80677
## 11 1.986008e-24 1.0000000 1.00000000  5.276241e+00 -2754.96989
## 12 1.909912e-24 1.0000000 1.00000000  6.001232e+00 -2755.12733
## 13 1.882985e-24 1.0000000 1.00000000  7.550050e+00 -2754.04851
## 14 1.852189e-24 1.0000000 1.00000000  9.034064e+00 -2752.19783
## 15 1.812038e-24 1.0000000 1.00000000  1.036132e+01 -2751.54750
## 16 1.773724e-24 1.0000000 1.00000000  1.171935e+01 -2751.02018
## 17 1.751011e-24 1.0000000 1.00000000  1.333878e+01 -2749.23957
## 18 1.743193e-24 1.0000000 1.00000000  1.520780e+01 -2747.50678
## 19 1.732688e-24 1.0000000 1.00000000  1.703178e+01 -2745.69437
## 20 1.730916e-24 1.0000000 1.00000000  1.900209e+01 -2744.11298
## 21 0.000000e+00 1.0000000 1.00000000 -8.000000e+00 -2742.13145
##            SBCp       PRESSp
## 1            NA           NA
## 2      1.901768 7.860936e+03
## 3   -206.049177 1.162523e+02
## 4   -241.357675 6.421632e+01
## 5   -308.989512 1.432666e+01
## 6  -3134.334274 3.272777e-24
## 7  -3135.209278 3.704628e-24
## 8  -3139.578024 3.842165e-24
## 9  -3140.156711 4.025125e-24
## 10 -3138.363739 4.132936e-24
## 11 -3135.628665 4.175185e-24
## 12 -3133.670106 4.121097e-24
## 13 -3130.468043 4.276321e-24
## 14 -3127.380507 4.618150e-24
## 15 -3124.564284 4.240605e-24
## 16 -3121.720817 4.716424e-24
## 17 -3118.453196 5.021762e-24
## 18 -3114.764900 5.107588e-24
## 19 -3111.155107 4.962129e-24
## 20 -3107.294245 5.262518e-24
## 21         -Inf 5.787842e-24